Abstract

Let be a singular nonintegral operator; that is, it does not have an integral representation by a kernel with size estimates, even rough. In this paper, we consider the boundedness of commutators with and Lipschitz functions. Applications include spectral multipliers of self-adjoint, positive operators, Riesz transforms of second-order divergence form operators, and fractional power of elliptic operators.

1. Introduction

Let be a bounded operator on for some , . A measurable function is called an associated kernel of if
holds for each continuous function with compact support and for almost all not in the support of .

The kernel is said to satisfy the following.

(i) The pointwise Hörmander condition on variable if there exist and such that
when , and denotes the ball with center , radius .

(ii) The integral Hörmander condition on variable if there exist constants and such that
for all .

It is well known that if is bounded on for some , , and , the two Hörmander conditions (i) and (ii) above are sufficient to imply that the commutator is bounded on for all , , with norm
where the commutator is defined by and is the BMO seminorm of . See [1, 2] for BMO functions on Euclidean spaces and [3] for spaces of homogeneous type.

A particular case of the result of Janson [2] states that is bounded, , if , . Here, is the homogeneous Lipschitz space determined by the first difference operator.

In [4], Duong and Yan have replaced the two Hörmander conditions (2) and (3) by the following weaker conditions (5) and (6) below which previously appeared in [5] and still concluded that the commutator is bounded on for all , . And in [6], Hu and Yang obtained the weighted boundedness of maximal commutator when satisfy (5) and (6). Roughly speaking, we assume the following.

(iii) There exists a class of operators with kernels , which satisfy the condition (23) in Section 2, so that the kernels of the operators satisfy the condition
when for some , , where is a positive constant.

(iv) There exists a class of operators with kernels , which satisfy the condition (23), such that have associated kernels and there exist positive constants , such that

Under conditions (5) and (6), if is bounded on for some , , then the commutator is bounded on for all , .

In [7], Auscher and Martell have considered the commutators of singular nonintegral operators, where the implicit terminology has been introduced in [8]. By this we mean that they are still of order 0, but they do not have an integral representation by a kernel with size and/or smoothness estimates. Let . Suppose that the singular nonintegral operator is a sublinear operator bounded on and that is a family of operators acting from into . Auscher and Martell assume the following.

(v) For all and all balls where denotes its radius,

(vi) For all and all balls where denotes its radius,

Let and (for the definitions of and see Section 2). Under conditions (7) and (8), if , then the commutator is bounded on ; that is, for all .

The main object of this paper is the commutators of nonintegral operators . Compared to the result in [7], we can obtain a more general result for belongs to the Lipschitz spaces . To be more specific, we can obtain the following.

Theorem 1. Let , such that . Suppose that is a sublinear operator bounded from to and that is a family of operators acting from into . Assume that
for all and all balls , where denotes its radius. Let such that . Let and . If , then there is a constant such that
for all and for all .

The case is understood in the sense that the -average in (10) is indeed an essential supremum.

Remark 2. Let be such that . Under the assumptions above, we know that if , then is bounded from to . See Theorem 2.2 in [9].In the limiting case , from the assumptions (9) and (10), we deduce
Consequently, from the Theorem 3.7 in [7], we know that if , then for and for all .

Theorem 3. Let . Suppose that is a sublinear operator bounded on and that is a family of operators acting from to . Assume that satisfy (9) and (10) with . Let , , and . Assume that there exists a constant such that . If , then there is a constant such that
for all .

2. Definitions and Preliminary Results

We use the notation
and we often ignore the Lebesgue measure and the variable of the integrand in writing integrals, unless this is needed to avoid confusions.

A weight is a nonnegative locally integrable function. We say that , , if there exists a constant such that for every ball
For , we say that if there is a constant such that for every ball , , for a.e. , or, equivalently, a.e., where denotes the classical Hardy-Littlewood maximal function of . The reverse Hölder classes are defined in the following way: , , if there is a constant such that for every ball

The endpoint is given by the condition: whenever, for any ball ,

The homogenous Lipschitz function space is the space of functions such that
where denotes the th difference operator (see [10]). That is, , , .

We have the following lemmas.

Lemma 4 (see [10]). For , , one has
For , the last formula should be modified appropriately.

Theorem A (see [7]). Fix , , and , . Then, there exist and with the following property: assume that , , , and are nonnegative measurable functions on such that for any cube there exist nonnegative functions and with for a.e. and
Then for all , and
As a consequence, for all , one has
provided , and
provided . Furthermore, if , then (24) and (25) hold, provided (whether or not ).

For and , we denote
where the supremum is taken with respect to all balls of positive measure containing the point .

Theorem B. Let , , and let and be the weight functions. For a constant to exist so that the inequality
would hold, it is necessary and sufficient that the condition
where , be fulfilled.

3. The Proof of the Main Theorems

Lemma 9. Let , , and . Let be a sublinear operator bounded from to .(i) If and , then .(ii) Assume that for any and for any one has that
where does not depend on and . Then for all , (33) holds.

Proof. The ideas of the following argument are taken from [7].Fix . Note that (i) follows easily observing that
since , imply that and hence, by assumption, .To obtain (ii), we fix and . Let be a cube such that . We may assume that since otherwise we can work with and observe that
Note that for , we have that and are finite almost everywhere since they belong to .Let and define as follows:
Then, it is immediate to see that for all . Thus, . As , we can use (33) and
To conclude, by Fatou’s lemma, it suffices to show that for a.e. and for some subsequence such that .As , for any , the dominated convergence theorem yields that in as . Therefore, is bounded from to . It follows that in . Thus, there exists a subsequence such that for a.e. . In this way we obtain
as desired, and we get that for a.e .

Proof of Theorem 1. We assume that , for , and the main ideas are the same and details are left to the interested reader. Lemma 9 ensures that it suffices to consider the case . Let and set . Note that by (i) of Lemma 9. Given a ball , we set and decompose as follows:
We observe that , where
and .We first estimate the average of on . Fix any . Let . Using Lemma 4,
Using (9) and Lemmas 4 and 5,
since . Hence, for any ,
We next estimate the average of on with . Using (10) and proceeding as before, we see that
for any . Thus we have obtained
For and , we can find a such that and . As mentioned before . Applying Theorem A and Remark 2 with in place of , we obtain
where we have used Lemma 6. This implies that

Proof of Theorem 3. Let , , and be the same as those used in the proof of Theorem 1. As mentioned before . Since , applying Theorem A with in place of and , we obtain
Noting that , Lemma 8 and Remark 2 give us that
This implies that

4. Applications

4.1. Spectral Multipliers: Off-Diagonal Estimates

Suppose that is a self-adjoint nonnegative definite operator on . Let be the spectral resolution of . For any bounded Borel function , by using the spectral theorem, we can define the operator
This is of course bounded on .

The following will be assumed throughout this subsection.(H1) is a nonnegative self-adjoint operator on .(H2) The operator generates an analytic semigroup which satisfies the Davies-Gaffney condition. That is, there exist constants such that for any open subsets ,
for every with , , where .(H3) Suppose . Assume that the analytic semigroup generated by satisfies “ off-diagonal” estimates: there exist coefficients satisfying such that for all balls and for all functions

Let be a nonnegative function such that
For , let denote the integer part of . Recall that is the space of functions on for which
is finite.

Then the following result holds.

Theorem 10. Let satisfy assumptions (H1)–(H3). Let be a nonnegative function satisfying (54), and suppose that the bounded measurable function satisfies
for some . Then(i) let . If and , then there is a constant such that
for all and for all .(ii) Let , , and . If there exists a constant such that , then there is a constant such that
for all and for all .

Proof. Estimate (57) follows from Theorem 1 with and estimate (58) follows from Theorem 3, applied to and with and . It suffices to show that there exist coefficients satisfying such that (9) and (10) hold for all .Fix . From (53), we deduce that
This estimate with in place of yields (10). Since, by functional calculus, , (9) was proved in [13].

4.2. Riesz Transforms

Let be an matrix of complex and -valued coefficients on . We assume that this matrix satisfies the following ellipticity (or “accretivity”) condition: there exist such that
for all and almost every . Associated with this matrix we define the second-order divergence form operator

The Riesz transforms associated to are , . Set . The solution of the Kato conjecture [14] implies that this operator extends boundedly to . This allows the representation
in which the integral converges strongly in both at and when .

Define by

We write for , .

We extract from [15] some definitions and results on unweighted off-diagonal estimates.

Definition 11. Let . One says that a family of sublinear operators satisfies full off-diagonal estimates, in short , if for some , for all closed sets and , all , and all , we have

If is a subinterval of , Int denotes the interior in of .

Proposition 12 (see [15]). Fix and .(a)There exists a nonempty maximal interval in , denoted by , such that if with , then satisfies full off-diagonal estimates and is a bounded set in .(b)There exists a nonempty maximal interval in , denoted by , such that if with , then satisfies full off-diagonal estimates and is a bounded set in .(c) and, for , we have if and only if .(d)Denote by , the lower and upper bounds of and by , those of . We have and . (We have set , the Sobolev exponent of when and , otherwise.)(e)If , . If , and with . (f)If , , , and .

Then for , satisfy (9) and (10) with and , where is a large enough integer. For the proof of this argument, see [15]. So Theorem 1 with and Theorem 3 can be applied to .

4.3. Fractional Operators

Let . The fractional power of an elliptic operator on is given formally by
with . There exist and , such that the semigroup is uniformly bounded on for every (see Proposition 12). We have the following results.

Lemma 13 (see [9]). Let so that . Fix a ball with radius . For and large enough, one has
and for
where and .

Theorem 14. Let , , and . Given , one has

Proof. We are going to apply Theorem 1 to the linear operator . We fix , , and so that . Then we can find , , such that , , and . Notice that as , we have that . By Theorem 1.2 in [9], we know that is bounded from to .We take , where is an integer to be chosen. We apply Lemma 13. Note that (66) is (9). Also, (10) follows from (67) after expanding . Then, we have that for by choosing . Consequently applying Theorem 1, we conclude that .

Acknowledgments

The authors would like to thank the referee for carefully reading the paper and for making several useful suggestions. This research was supported by Tianyuan Fund for Mathematics (Grant no. 11226100), Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20124410120002), and SRF of Guangzhou Education Bureau (Grant no. 2012A088).